While staying in a 15-story hotel, Polya plays the following game. She enters an elevator on the $6th$ floor. She flips a fair coin five times to determine her next five stops. Each time she flips heads, she goes up one floor. Each time she flips tails, she goes down one floor. What is the probability that each of her next five stops is on the $7th$ floor or higher? Express your answer as a common fraction.